Fractals, baby! Vote now!

For the discussion of math. Duh.

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Which is your "favorite"?

Koch Snowflake
25
6%
Sierpiński Triangle
33
8%
Sierpiński Carpet
7
2%
Cantor Set
23
6%
Menger Sponge
17
4%
Dragon Curve
34
8%
Romanesco Broccoli
48
12%
Mandelbrot Set
81
20%
Peano Curve
2
0%
Space-filling Curve
21
5%
Levy Flight
4
1%
Kleinian Group
9
2%
Julia Set
33
8%
Lyapunov fractal
23
6%
T-Square
3
1%
Brownian Tree
6
1%
Phoenix Set
15
4%
Otter/Duck Set
18
4%
 
Total votes: 402

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GCM
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Fractals, baby! Vote now!

Postby GCM » Thu May 08, 2008 1:12 am UTC

Please let me know if there's anything I've missed so I can totally ignore you! :D Because I can't do anything about it now.
I'm kinda new to this whole thing, so please don't berate me yet. I've gotten these off the web, and most of my though now goes to "they're very pretty". So do vote. Based on that. In fact, I'm not supposed to be even looking at this yet: Teach = "Dammit, Chris, you're not supposed to see this till college! Stop wasting your time on this and do your homework! I want that A, and I'm not going to get it if you don't freakin' study!"
Last edited by GCM on Thu May 08, 2008 12:05 pm UTC, edited 1 time in total.
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Re: Fractals, baby! Vote now!

Postby 3.14159265... » Thu May 08, 2008 1:24 am UTC

I think you should provide pictures.
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Re: Fractals, baby! Vote now!

Postby Qoppa » Thu May 08, 2008 2:22 am UTC

3.14159265... wrote:I think you should provide pictures.
+1

I don't feel like wikiing everything but the Mandelbrot set.

Code: Select all

_=0,w=-1,(*t)(int,int);a()??<char*p="[gd\
~/d~/\\b\x7F\177l*~/~djal{x}h!\005h";(++w
<033)?(putchar((*t)(w??(p:>,w?_:0XD)),a()
):0;%>O(x,l)??<_='['/7;{return!(x%(_-11))
?x??'l:x^(1+ ++l);}??>main(){t=&O;w=a();}

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Re: Fractals, baby! Vote now!

Postby 3.14159265... » Thu May 08, 2008 4:50 am UTC

Koch Snowflake
Image
Sierpiński Triangle
Image
Sierpiński Carpet
Image
Cantor Set
Image
Menger Sponge
Image
Dragon Curve
Image
Romanesco Broccoli
Image
Mandelbrot Set
Image
Peano Curve and Space filling Curve
Image
Levy Flight
Image
Kleinian Group
Image
Julia Set
Image
Lyapunov fractal
Image
T-Square
Image
Brownian Tree
Image
Phoenix Set
Image
Otter
Image
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Re: Fractals, baby! Vote now!

Postby Silas » Thu May 08, 2008 5:00 am UTC

Aren't "space-filling curves" a category of fractals, not a specific one? And why isn't the Hilbert Curve on the list?

(The Sierpinski Gasket makes me want to cry.)
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Re: Fractals, baby! Vote now!

Postby skeptical scientist » Thu May 08, 2008 5:20 am UTC

I voted for space filling curves - they aren't as pretty as some of the others, but the fact that they exist is just too cool.

The Cantor set was a close second. Interesting fact: every perfect, compact, totally disconnected (nonempty) metric space is homeomorphic to the Cantor set. It also makes a fun exercise.
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Re: Fractals, baby! Vote now!

Postby Silas » Thu May 08, 2008 6:56 am UTC

I just looked up the Romanesco Broccoli. Imagine my surprise when I learned it's an actual broccoli- the edible flower of a sort of mustard plant. You know, the kind that grows in a field, and has leaves and roots and pollen. For gathering sunlight and water and whatnot. My mind is blown.
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Re: Fractals, baby! Vote now!

Postby ++$_ » Thu May 08, 2008 7:00 am UTC

I voted for Mandelbrot, the dragon curve, and the Cantor set. Because you can't not vote for the Mandelbrot set, the dragon curve is just awesome, and the Cantor set is seriously weird.

skeptical scientist wrote:I voted for space filling curves - they aren't as pretty as some of the others, but the fact that they exist is just too cool.

The Cantor set was a close second. Interesting fact: every perfect, compact, totally disconnected (nonempty) metric space is homeomorphic to the Cantor set. It also makes a fun exercise.
You get 3 votes, so you can vote for both if you want.

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Re: Fractals, baby! Vote now!

Postby Robin S » Thu May 08, 2008 10:26 am UTC

Mandelbrot and otter/duck: the Mandelbrot set contains approximate images of every Julia set in its family, and is itself a subset of a higher-dimensional fractal with various pretty cross-sections.
This is a placeholder until I think of something more creative to put here.

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Re: Fractals, baby! Vote now!

Postby Torn Apart By Dingos » Thu May 08, 2008 10:28 am UTC

I love spacefilling curves! I'll use this thread to show an image I made. I mapped RGB (three-dimensional) to the unit square (2D dimensional) with a three- and a two-dimensional Sierpinski curve. I tried it with Hilbert and Peano curves as well, and I also tried using the HSV color space, but this one was the most pretty and symmetric.

Image

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Re: Fractals, baby! Vote now!

Postby Dobblesworth » Thu May 08, 2008 11:26 am UTC

I put one vote down for Mandelbrot Set, as you can never deny the all-consuming awesomeness of a Rorscach Test on Fire. My others went to Sierpinski Triangle (or Triforce9000) and Koch Snowflake.

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Re: Fractals, baby! Vote now!

Postby GCM » Thu May 08, 2008 12:02 pm UTC

Grates to π for the images; I really appreciate it. It's something I won't overlook next time.

Silas wrote:Aren't "space-filling curves" a category of fractals, not a specific one? And why isn't the Hilbert Curve on the list?


Point taken, but I've not studied them before, so voila.
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Re: Fractals, baby! Vote now!

Postby tiny » Thu May 08, 2008 2:28 pm UTC

The cool broccoli (which is a hybrid of broccoli and cauliflower), Mandelbrot and Phoenix are my favourite.
The Sierpinski Carpet makes me feel strangely empty inside O.o
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Re: Fractals, baby! Vote now!

Postby Citizen K » Thu May 08, 2008 2:56 pm UTC

I cast an Otter vote for an unavailable option: the classic fern fractal they always have you make in math/programming courses.
Rather large picture from Wikipedia in spoiler:
Spoiler:
Image

I just find it very aesthetically pleasing. Pretty, and sort of minimalist compared to the flashier ones like the Mandelbrot set. (Which also got a vote, along with the triangle)

Tiny wrote:The Sierpinski Carpet makes me feel strangely empty inside O.o

I agree. Double for that Menger sponge. Although it does get a few points for looking reminiscent of a Borg cube.
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Re: Fractals, baby! Vote now!

Postby Frimble » Thu May 08, 2008 5:15 pm UTC

How about Quaternion Julia fractals? Hard to appreciate, as they have fractal dimensions between 3 and 4 but quite pretty anyway.

(please correct me if I have got my terminology wrong, I have had trouble finding good explanations on the subject)
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Re: Fractals, baby! Vote now!

Postby Robin S » Thu May 08, 2008 5:20 pm UTC

They're still Julia sets, just plotted over a larger domain.
This is a placeholder until I think of something more creative to put here.

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Re: Fractals, baby! Vote now!

Postby Various Varieties » Thu May 08, 2008 5:32 pm UTC


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Re: Fractals, baby! Vote now!

Postby Frimble » Thu May 08, 2008 5:47 pm UTC

Robin S wrote:They're still Julia sets, just plotted over a larger domain.


Larger domain? In what sense? There are the same number of points in a 4 dimensional space as there are in a 2 dimensional space. (I'm sorry I can't help it :) )

In any case doesn't the fact that their domain is a non-abelian (non-commutative) group entitle them to their own category?

Does this Count


I wouldn't have thought so, Britain's coastline may be complicated, but it is not infinite. Even if you include the length of every last stream and drain that ends in the sea the it still has a dimensions of Length as opposed to Length^x.

PS. I question romanesco broccoli's claim to being a fractal too.
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Re: Fractals, baby! Vote now!

Postby Robin S » Thu May 08, 2008 5:52 pm UTC

Frimble wrote:In any case doesn't the fact that their domain is a non-abelian (non-commutative) group entitle them to their own category?
I don't see why. Last time I checked, Julia sets were defined as sets of points for which behaviour under an iterated function was chaotic. Whether the domain happens to commute under multiplication is irrelevant.

As for the coastline / broccoli question, I would suggest that the inclusion of the latter in the poll would suggest that the former was legitimate. If you're questioning the validity of the inclusion of the latter, you could take it up with the original poster, but I would point out that none of the images are of the fractals themselves, but merely approximations to them - indeed, we are incapable of directly visualizing the fractals themselves, so that level of pedantry renders the poll useless.
This is a placeholder until I think of something more creative to put here.

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Re: Fractals, baby! Vote now!

Postby Frimble » Thu May 08, 2008 7:03 pm UTC

The Oxford Concise Dictionary of Mathematics defines a Julia set as: 'The boundary of the set of points z0 in the complex plane for which the application of the function f(z)=z2+c repeatedly to the point z0 produces a bounded sequence. The term may be used similarly for other functions as well.'

As this definition refers to the complex plane, it does not include quaternion Julia's.

Robin S wrote:
As for the coastline / broccoli question, I would suggest that the inclusion of the latter in the poll would suggest that the former was legitimate. If you're questioning the validity of the inclusion of the latter, you could take it up with the original poster, but I would point out that none of the images are of the fractals themselves, but merely approximations to them - indeed, we are incapable of directly visualizing the fractals themselves, so that level of pedantry renders the poll useless.


It is my belief that a concept can be beautiful regardless of whether we can visualise it. After all who can visualise music? On second thoughts my debating skills are not up to arguing this... I had probably better stop now...

Alright the broccoli's an approximation to a fractal because it has a shape which if 'perfected' could potentially have an infinite surface area in a finite volume.

But by what argument does a coastline have an infinite length in a finite area?
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Re: Fractals, baby! Vote now!

Postby Bassoon » Thu May 08, 2008 8:41 pm UTC

Phoenix Fractal FTW!

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Re: Fractals, baby! Vote now!

Postby Robin S » Thu May 08, 2008 10:51 pm UTC

Frimble wrote:The Oxford Concise Dictionary of Mathematics defines a Julia set as: 'The boundary of the set of points z0 in the complex plane for which the application of the function f(z)=z2+c repeatedly to the point z0 produces a bounded sequence. The term may be used similarly for other functions as well.'

As this definition refers to the complex plane, it does not include quaternion Julia's.
"Other functions" can have other domains. A quaternion Julia set is just a Julia set which happens to have been defined over the quaternions. Julia sets generated by the mapping z -> z2+c are the best-known family in this category, and often the term "Julia set" applies to that family (and hence is restricted to the complex plane), but not necessarily.

It is my belief that a concept can be beautiful regardless of whether we can visualise it. After all who can visualise music? On second thoughts my debating skills are not up to arguing this... I had probably better stop now...
I agree with you that things can be beautiful for reasons other than how they look. However, the main reason I (and, I think, many people) find fractals beautiful is because of how they look.

Alright the broccoli's an approximation to a fractal because it has a shape which if 'perfected' could potentially have an infinite surface area in a finite volume.

But by what argument does a coastline have an infinite length in a finite area?
By the same argument.
This is a placeholder until I think of something more creative to put here.

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Re: Fractals, baby! Vote now!

Postby Luthen » Fri May 09, 2008 4:49 am UTC

Dragon curve: duh ==>
I like it cause I can make it manually.

Mandelbrot is the poster child but I really only like the zoomed in sections (which shouldn't really make a difference should it?)

Never seen the Lyapunov before but it was just so insane it got my vote.

Would like to join the discussion but don't know enough of the maths behind them (did the Koch Snowflake for a friend's math assignment last year).
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Re: Fractals, baby! Vote now!

Postby G Mess » Fri May 09, 2008 5:36 am UTC

Do I see the Burning Ship fractal?

I don't think I do.

This makes me sad, but the romanesco ______ is just slightly less cool, so it got my vote.

But damn, dude!

Burning ships.

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Re: Fractals, baby! Vote now!

Postby ++$_ » Fri May 09, 2008 7:00 am UTC

Robin S wrote:
Alright the broccoli's an approximation to a fractal because it has a shape which if 'perfected' could potentially have an infinite surface area in a finite volume.

But by what argument does a coastline have an infinite length in a finite area?
By the same argument.
How would one "perfect" a coastline? With the broccoli, there seems to be a pattern that one could follow. Not so much with the coast of England.

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Re: Fractals, baby! Vote now!

Postby Robin S » Fri May 09, 2008 10:42 am UTC

Using something like a fractal terrain generator.
This is a placeholder until I think of something more creative to put here.

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Re: Fractals, baby! Vote now!

Postby BeetlesBane » Fri May 09, 2008 4:21 pm UTC

From the definition of fractal at http://mathworld.wolfram.com/Fractal.html
"The prototypical example for a fractal is the length of a coastline measured with different length rulers."

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Re: Fractals, baby! Vote now!

Postby Frimble » Fri May 09, 2008 6:40 pm UTC

Robin S wrote:
Frimble wrote:The Oxford Concise Dictionary of Mathematics defines a Julia set as: 'The boundary of the set of points z0 in the complex plane for which the application of the function f(z)=z2+c repeatedly to the point z0 produces a bounded sequence. The term may be used similarly for other functions as well.'

As this definition refers to the complex plane, it does not include quaternion Julia's.
"Other functions" can have other domains. A quaternion Julia set is just a Julia set which happens to have been defined over the quaternions. Julia sets generated by the mapping z -> z2+c are the best-known family in this category, and often the term "Julia set" applies to that family (and hence is restricted to the complex plane), but not necessarily.


Alright, The Julia set gets my vote then.

Robin S wrote:Using something like a fractal terrain generator.


A fractal terrain generator? Could you give an example? I have a program that can turn Julia/phoenix/mandlebrot fractals into 3d models, but I wouldn't call this a perfection of a coastline.

BeetlesBane wrote:From the definition of fractal at http://mathworld.wolfram.com/Fractal.html
"The prototypical example for a fractal is the length of a coastline measured with different length rulers."


The oxford dictionary defines a fractal in terms of its fractal dimension rather than in terms of symmetry. I'm not sure which is correct or even if the definitions are equivalent. The oxford definition seems less ambiguous in any case.
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Re: Fractals, baby! Vote now!

Postby The Hyphenator » Sat May 10, 2008 5:57 pm UTC

Mandelbrot set. That is the prettiest and most amazing fractal ever. Nothing else comes close, in my opinion (except for the otter/duck set, of course).
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Spoiler:
Image

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Re: Fractals, baby! Vote now!

Postby thornahawk » Mon Nov 17, 2008 6:10 pm UTC

I like Menger's sponge:

Code: Select all

(* If[$VersionNumber < 5] *)
Show[Graphics3D[
    Join[{EdgeForm[], SurfaceColor[Hue[0.47], Hue[0.12], 1.38]},
      Nest[Function[
          g3d, (Flatten[
                    Cases[g3d, _Cuboid, Infinity]] /. {x_?NumericQ,
                      y_?NumericQ, z_?NumericQ} :> {x, y, z}/3 - #) & /@
            Select[Flatten[Outer[List, Sequence @@ Table[2{-1, 0, 1}/3, {3}]],
                 2], (Count[#, 0] < 2) &]], {Cuboid[{-1, -1, -1}, {1, 1, 1}]},
         4]], Boxed -> False]]
(* If[$VersionNumber >= 5] *)
Show[Graphics3D[
    Join[{EdgeForm[], SurfaceColor[Hue[0.47], Hue[0.12], 1.38]},
      Nest[Function[
          g3d, (Flatten[
                    Cases[g3d, _Cuboid, Infinity]] /. {x_?NumericQ,
                      y_?NumericQ, z_?NumericQ} :> {x, y, z}/3 - #) & /@
            Select[Tuples[{-2/3, 0, 2/3},
                3], (Count[#, 0] < 2) &]], {Cuboid[{-1, -1, -1}, {1, 1, 1}]},
        2]], Boxed -> False]]


but the pentaflaked dodecahedron remains the fractal I love the most.

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Re: Fractals, baby! Vote now!

Postby atimholt » Tue Nov 25, 2008 9:41 pm UTC

I like this "clockwork arm" curve I came up with but haven't been able to find on the internet:
x = Sum[cos(t * 2^n) / 2^n | n ranges from 0 to infinity]
y = Sum[sin(t * 2^n) / 2^n | n ranges from 0 to infinity]
and t ranges from 0 to 2pi (of course)

Just imagine, on the first iteration, you have a pencil on the end of a rotating arm. The arm completes a rotation, drawing a circle
On the second iteration, you add another arm onto the end, one that is half as long and rotates twice as fast (in relation to the stationary drawing paper, not the first arm: I tried that and, to me, it doesn't look as cool.)
The third iteration adds a third arm one eighth the length of the original and rotating eight times as fast.
I might download a crappy free graphing program and generate a picture to show you.

It looks a little like an alien's head.

EDIT: Actually, that was really quick and easy. Here's eight iterations (surely overkill):
Spoiler:
Image

And a zoom in:
Spoiler:
Image
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Re: Fractals, baby! Vote now!

Postby thornahawk » Wed Nov 26, 2008 7:02 am UTC

atimholt wrote:I like this "clockwork arm" curve I came up with but haven't been able to find on the internet:
x = Sum[cos(t * 2^n) / 2^n | n ranges from 0 to infinity]
y = Sum[sin(t * 2^n) / 2^n | n ranges from 0 to infinity]
and t ranges from 0 to 2pi (of course)


Spoiler:

Code: Select all

ParametricPlot[{NSum[(2^-k)Cos[t 2^k], {k, 0, Infinity}, Method -> SequenceLimit],
    NSum[(2^-k)Sin[t 2^k], {k, 0, Infinity}, Method -> SequenceLimit]}, {t, 0, 2 Pi},
    AspectRatio -> Automatic, Axes -> None, Frame -> True, PlotPoints -> 75]


Image


If it helps anything, the functions you used bear a resemblance to the Fourier series studied by Weierstrass and Riemann.

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Re: Fractals, baby! Vote now!

Postby sh3l1 » Sat Nov 29, 2008 12:23 am UTC

I voted for space filling, because they are useful. Mandelbrot for prettiness.

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Re: Fractals, baby! Vote now!

Postby DrProfessorPhD » Sun Nov 30, 2008 12:50 am UTC

Mandelbrot, Phoenix, and Otter (Burning ship)
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Re: Fractals, baby! Vote now!

Postby taby » Tue Dec 23, 2008 4:25 pm UTC

Hi Frimble,

I have an image of a "tornado-lobster" quaternion Julia set. I thought you might be interested. :)

It is Figure 14, on page 17 of this paper: http://cavekitty.ca/inv_ssa.pdf

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Re: Fractals, baby! Vote now!

Postby Xutar » Wed Dec 24, 2008 12:32 am UTC

How could you leave out http://en.wikipedia.org/wiki/Burning_Ship_fractal?
(I voted mandelbrot since they are closely related)

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Re: Fractals, baby! Vote now!

Postby parallax » Wed Dec 24, 2008 1:00 am UTC

The coastline of Britain is a fractal. It has a fractal dimension of 1.25. Granted, it may not be an exact fractal as the coastline itself is poorly defined at smaller scales, but it is statistically self-similar at all larger scales.
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Re: Fractals, baby! Vote now!

Postby rockym93 » Wed Dec 24, 2008 3:08 am UTC

I voted Mandelbrot set and Dragon curve.

Mandelbrot set 'cause I know it inside out because of a project I had to do on it (set of bounded numbers on the complex plane... forgotten the formula) and the Dragon curve because iterations of it appear on the chapter pages of Jurassic Park, which is just so awesome.
"It's kinda fun to do the impossible" - Walt Disney

Rentsy
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Re: Fractals, baby! Vote now!

Postby Rentsy » Thu Dec 25, 2008 4:23 am UTC

Pathological monsters! Every one of them is a splinter in my eye!

The lyapunov is all... curvaceous. Even better, I think I understand the math behind all the pretty curves.

[edit] The coastline of Britain is infinitely long.

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roundedge
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Re: Fractals, baby! Vote now!

Postby roundedge » Sat Dec 27, 2008 9:00 am UTC

if anyone can find me a fractal with a Hausdroff dimension of π, that would be my favourite.


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